Functional equations and distribution functions
Let \(a \in (0,1)\), \(N \in \mathbb{N} \backslash \{1\}\) and \(- 1 = \beta_0 \leq \beta_1 \leq \dots \leq \beta_{N - 1} = 1\). Then the functional equation \[ f(x) = {1 \over N} \sum^{N - 1}_{k = 0} f \left( {x - \beta_k \over a} \right) \] has a unique bounded solution \(f : \mathbb{R} \to \mathbb{R}\) vanishing on \((- \infty, -1/(1 - a))\) and taking the value 1 on \((1/(1 - a), + \infty)\). This unique solution \(f\) is continuous and increasing. It is strictly increasing on \([- 1/(1 - a), 1/(1 - a)]\) iff \(2a/(1 - a) \geq \max \{\beta_k - \beta_{k - 1} : 1 \leq k \leq N - 1\}\). If \(2a/(1 - a) < \min \{\beta_k - \beta_{k - 1} : 1 \leq k \leq N - 1\}\) then \(f\) is singular. It is also singular in the case where \(\beta_k = - 1 + 2k/(N - 1)\) for \(k \in \{0, \dots, N - 1\}\) and \(a\) is the inverse of a nonrational Pisot number. If \(\beta_k = - 1 + 2k/(N - 1)\) for \(k \in \{0, \dots, N - 1\}\) and \(a = N^{- 1/p}\) with an integer \(p\), then \(f\) is absolutely continuous, has a \((p - 1)st\) derivative which is a continuous and piecewise linear function (and a formula for \(f\) is given via convolutions). An interesting connection with the Schilling equation \[ 4q \varphi (qx) = \varphi (x - 1) + \varphi (x + 1) + 2 \varphi (x), \] where \(q \in (0,1)\) is a fixed parameter, is also pointed out.
singular and absolutely continuous functions
scientific article; zbMATH DE number 785165